Problem 296 - Elementary Algebra

Question

In the following exercises, simplify. $$ (4+9 \sqrt{3})(4-9 \sqrt{3}) $$

Step-by-Step Solution

Answer

-227

Step 1: Identify the Expression

The given expression to simplify is \((4 + 9 \sqrt{3})(4 - 9 \sqrt{3})\).

Step 2: Recognize the Difference of Squares

This expression is in the form of \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 4\) and \(b = 9 \sqrt{3}\).

Step 3: Apply the Difference of Squares Formula

Using the formula, we get \((4 + 9 \sqrt{3})(4 - 9 \sqrt{3}) = 4^2 - (9\sqrt{3})^2\).

Step 4: Simplify the Expression

Calculate the squares: \(4^2 = 16\) and \((9 \sqrt{3})^2 = 81 \times 3 = 243\).

Step 5: Subtract the Values

Subtract the squared values: \(16 - 243 = -227\).

Key Concepts

difference of squares

The expression \( (4 + 9 \sqrt{3})(4 - 9 \sqrt{3}) \) is a classic example of the difference of squares.
The difference of squares is a special factoring technique and one of the most useful algebraic identities.

It can be recognized when you have two terms being multiplied, one with a plus and one with a minus, like \( (a + b)(a - b) \).
This always simplifies to \( a^2 - b^2 \).
In our example:

• \textbf{a} = 4
• \textbf{b} = 9 \sqrt{3}

mathematical formulas

Recognizing and applying the right mathematical formula is key to simplifying expressions.
Here, we use the formula for the difference of squares: \[ (a + b)(a - b) = a^2 - b^2 \].
In the given problem, by substituting \ a \ and \ b \ into the formula, you ensure that the expression is correctly simplified.
This formula helps transform and reduce complex expressions into more manageable numbers or simpler forms.
Understanding how and when to apply different mathematical formulas will make solving algebra problems much easier.

simplifying expressions

Simplifying expressions can sometimes feel tricky, but breaking them into smaller steps can help.
Using known algebraic formulas and properties, such as difference of squares, makes this easier.
For example, simplifying \( (4 + 9 \sqrt{3})(4 - 9 \sqrt{3}) \) involves recognizing the pattern and applying the correct formula.
By calculating step by step: First square each part: \( 4^2 \) and \( (9 \sqrt{3})^2 \), then subtract: \ 16 - 243 = -227 \. As a result, you get the final simplified value.

square roots

Square roots often appear in algebra problems, and it’s important to know how to deal with them.
In our problem, \( 9 \sqrt{3} \) is involved.
When square roots are squared, they simplify to regular numbers.
So, \: \ (9 \sqrt{3})^2 = 81 \times 3 = 243 \. This step is crucial in following the steps correctly.
Knowing these properties can help simplify the expressions quickly and correctly when working with square roots.

subtraction

In the final step of simplifying, subtraction is used to find the difference.
After squaring the terms in the expression \( 4^2 - (9 \sqrt{3})^2 \), you get: \ 16 - 243 \.
It's important to carefully subtract to avoid mistakes.
Here, \ 16 - 243 \ results in \ -227 \.
Subtraction helps to obtain the final value of the simplified expression.
Ensuring accuracy in this step ensures that the solution is correct.